I assume you are referring to Elrond and Elros. That was a Valar directive after the end if the First Age. Elros still had a long life, but he was counted among Men, which meant when he died his soul went where ever Mens souls go. And no one knows except Iluvatar.

Elves don't really get a choice, it was just that occasion when there were half elves, decendants of Beren (man) and Luthien (elf), and really just Elrond and Elros. All the other people born from human and elf relations are not really mentioned.

Elvish immortality is not immortality as we normally see it. Elves are part of Middle Earth, literally. They can die in the same way humans can, wounds, disease and grief. When they do, they end up back at Valinor in the halls of Mandos. They can, and usually do, come back out and live in Valinor with the rest of the Valar. They can never leave like humans do.

This also applies to Elladan, Elrohir, and Arwen as well. As Elrond's children, they were also given the same choice as he was, though they were able to defer it for an indefinite period it would seem. It's unclear what Elladan and Elrohir ultimately chose, but we know Arwen's choice.